Simplify the following expression and state the condition under which the simplification is valid. $x = \dfrac{3p^2 + 57p + 270}{5p^2 + 100p + 500}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ x = \dfrac {3(p^2 + 19p + 90)} {5(p^2 + 20p + 100)} $ $ x = \dfrac{3}{5} \cdot \dfrac{p^2 + 19p + 90}{p^2 + 20p + 100} $ Next factor the numerator and denominator. $ x = \dfrac{3}{5} \cdot \dfrac{(p + 10)(p + 9)}{(p + 10)(p + 10)}$ Assuming $p \neq -10$ , we can cancel the $p + 10$ $ x = \dfrac{3}{5} \cdot \dfrac{p + 9}{p + 10}$ Therefore: $ x = \dfrac{ 3(p + 9)}{ 5(p + 10)}$, $p \neq -10$